License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CONCUR.2021.28
URN: urn:nbn:de:0030-drops-144053
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14405/
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Baier, Christel ; Funke, Florian ; Jantsch, Simon ; Karimov, Toghrul ; Lefaucheux, Engel ; Luca, Florian ; Ouaknine, Joël ; Purser, David ; Whiteland, Markus A. ; Worrell, James

The Orbit Problem for Parametric Linear Dynamical Systems

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LIPIcs-CONCUR-2021-28.pdf (0.8 MB)

Abstract

We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with two or more parameters.
More precisely, consider a d-dimensional square matrix M whose entries are algebraic functions in one or more real variables. Given initial and target vectors u,v ∈ ℚ^d, the parametric point-to-point orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix N ∈ ℝ^{d× d}, and a positive integer n ∈ ℕ, such that N^{n} u = v.
We show decidability for the case in which M depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting intractability in the case of two or more parameters.

BibTeX - Entry

@InProceedings{baier_et_al:LIPIcs.CONCUR.2021.28,
  author =	{Baier, Christel and Funke, Florian and Jantsch, Simon and Karimov, Toghrul and Lefaucheux, Engel and Luca, Florian and Ouaknine, Jo\"{e}l and Purser, David and Whiteland, Markus A. and Worrell, James},
  title =	{{The Orbit Problem for Parametric Linear Dynamical Systems}},
  booktitle =	{32nd International Conference on Concurrency Theory (CONCUR 2021)},
  pages =	{28:1--28:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-203-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{203},
  editor =	{Haddad, Serge and Varacca, Daniele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14405},
  URN =		{urn:nbn:de:0030-drops-144053},
  doi =		{10.4230/LIPIcs.CONCUR.2021.28},
  annote =	{Keywords: Orbit problem, parametric, linear dynamical systems}
}

Keywords: Orbit problem, parametric, linear dynamical systems
Collection: 32nd International Conference on Concurrency Theory (CONCUR 2021)
Issue Date: 2021
Date of publication: 13.08.2021

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